Mesh Not Moving Properly With Armature
You should unclick the Bone Envelopes option of the Armature modifier of the mesh, then select the armature, shift select the mesh, ctrl tab to switch to Weight Paint mode, then select each bone and paint the mesh accordingly to the influence you want this bone to have on the mesh.

Or even delete the Armature modifier and reparent the mesh to the armature With Automatic Weight and then bring some corrections in Weight Paint mode...